What are the angles of a right triangle?

A right triangle has one right angle of exactly 90 degrees and two acute angles. These two acute angles are always complementary, meaning that their sum is 90 degrees. The sum of all three angles therefore adds up to the usual 180 degrees.

How do you calculate the angles of a right triangle given two sides?

You take the ratio of two sides and apply the corresponding inverse trigonometric function. If you take the value of the opposite side divided by the adjacent side and use the inverse tangent function, you can directly calculate one of the acute angles. You can also use the inverse sine or cosine function if one of the known sides is the hypotenuse.

What are the angles of a 3-4-5 triangle?

The angles are approximately 36.87 degrees and 53.13 degrees, plus a right angle of 90 degrees. The leg with length 3 is opposite the smaller angle, while the leg with length 4 is opposite the larger angle.

Can a right triangle have two equal angles?

Yes, in this case both of the acute angles are each 45 degrees. This is an isosceles right triangle which is called a 45-45-90 triangle because it has two legs that are the same length. A right triangle cannot have two right angles since they would use up all 180 degrees.

Do you need the length of sides to determine an angle?

It is not strictly necessary. If you know one acute angle, the other can be found by subtracting this value from 90 degrees. To determine the actual length of a side, at least one side length must be known. However, if only angles are desired, then one measurement suffices since two angles are complementary.

Angle of a Right Triangle Calculator

Your right triangle

Side a (leg opposite angle alpha)

Side b (leg opposite angle beta)

Hypotenuse c (opposite the right angle)

Angle alpha (at vertex A)

Angle beta (at vertex B)

Enter any two values and the calculator completes the right triangle. Two sides give you both acute angles; one side plus one angle gives everything else.

Please input values into the fields

Good to know

To get an acute angle from two sides, use the inverse tangent:

alpha = arctan(a ÷ b)

. Give it one side and one angle instead and it switches to sine, cosine, or the Pythagorean theorem to find the rest.

Right-triangle angles show up whenever a slope, a ramp, a roof, or a line of sight meets level ground. Carpenters, surveyors, and students all reach for the same inverse-trig relationships this tool uses.

Any right triangle has one right angle of 90 degrees, and the remaining two acute angles make up the rest of the 180 degrees. This calculator will determine both of the acute angles as well as other information about the triangle given any two known values.

If you input the lengths of two sides it will give you both acute angles. If you specify a side and an angle then it can calculate the length of the remaining side, area, perimeter and even the height to hypotenuse.

Angle of a right triangle

Label the three sides of your triangle a, b and c, with c being the hypotenuse opposite the right angle. Label the acute angle opposite side a as α (alpha) and the acute angle opposite side b as β (beta).

The sum of the angles is determined by the right angle. The sum of alpha and beta is always 90 degrees. Therefore, if one acute angle is known, then the other can easily be calculated. For this reason, these two angles are called complementary angles.

\alpha + \beta = 90^\circ \qquad \beta = 90^\circ - \alpha

Calculating angles with two sides.

When two sides are known, an inverse trigonometric function can be used to convert this ratio into the angle. Which function is used depends on which two sides are known.

The tangent function uses the two legs of a right triangle; the sine function uses the opposite leg and hypotenuse; and the cosine function uses the adjacent leg and hypotenuse.

\alpha = \arctan\!\left(\frac{a}{b}\right) = \arcsin\!\left(\frac{a}{c}\right) = \arccos\!\left(\frac{b}{c}\right)

For angle β we use the same ratio but switch the positions of the two legs. In this case side b is the opposite leg.

\beta = \arctan\!\left(\frac{b}{a}\right) = \arcsin\!\left(\frac{b}{c}\right)

The table below uses a 3–4–5 triangle to illustrate the various parts of the triangle.

Symbol	Meaning	Example
a	Leg opposite angle alpha	3
b	Leg opposite angle beta	4
c	Hypotenuse, opposite the right angle	5
alpha	Acute angle facing side a	36.87 degrees
beta	Acute angle facing side b	53.13 degrees

A full example:

Consider the typical 3-4-5 triangle with legs of length 3 and 4. Since side a is 3 and side b is 4, then α can be directly calculated from the ratio of the tangent function.

\alpha = \arctan\!\left(\frac{3}{4}\right) = \arctan(0.75) \approx 36.87^\circ

Beta is the angle remaining after subtracting 90 degrees.

\beta = 90^\circ - 36.87^\circ = 53.13^\circ

The hypotenuse can be directly calculated using the Pythagorean theorem and the two shorter sides can also be used to determine area and height.

c = \sqrt{3^2 + 4^2} = 5 \qquad A = \tfrac{1}{2}(3)(4) = 6

Special right triangles.

In a 45-45-90 triangle, the two legs are equal in length to each other, so both acute angles are 45 degrees, and the hypotenuse is about 1.414 times as long as either leg.

The acute angles of a 30-60-90 triangle are 30 degrees and 60 degrees respectively, and the side lengths are in the ratio 1:√3:2. If your input values fit either of these two cases, then this calculator will indicate so.

If all three sides are integers (whole numbers), such as 3-4-5, 5-12-13 and 8-15-17, then it is a Pythagorean triple. The angles will not be simple values but will depend on the ratio of the sides.

This is how this calculator works:

Enter two known values and leave the remaining fields blank. The most common case is when you enter two sides; the tool will immediately calculate the two acute angles.

You can also enter a side and an acute angle. The tool uses sine, cosine and Pythagoras' theorem to calculate the other sides. The units for angles can be switched between degrees, radians and gradians, as well as the units for lengths. This is because the angles only depend on the ratio of the side lengths.

This tool is for learning and easy checking, when doing construction, manufacturing or surveying work you should always check angles with calibrated instruments before cutting or erecting.

Frequently asked questions

What are the angles of a right triangle?: A right triangle has one right angle of exactly 90 degrees and two acute angles. These two acute angles are always complementary, meaning that their sum is 90 degrees. The sum of all three angles therefore adds up to the usual 180 degrees.
How do you calculate the angles of a right triangle given two sides?: You take the ratio of two sides and apply the corresponding inverse trigonometric function. If you take the value of the opposite side divided by the adjacent side and use the inverse tangent function, you can directly calculate one of the acute angles. You can also use the inverse sine or cosine function if one of the known sides is the hypotenuse.
What are the angles of a 3-4-5 triangle?: The angles are approximately 36.87 degrees and 53.13 degrees, plus a right angle of 90 degrees. The leg with length 3 is opposite the smaller angle, while the leg with length 4 is opposite the larger angle.
Can a right triangle have two equal angles?: Yes, in this case both of the acute angles are each 45 degrees. This is an isosceles right triangle which is called a 45-45-90 triangle because it has two legs that are the same length. A right triangle cannot have two right angles since they would use up all 180 degrees.
Do you need the length of sides to determine an angle?: It is not strictly necessary. If you know one acute angle, the other can be found by subtracting this value from 90 degrees. To determine the actual length of a side, at least one side length must be known. However, if only angles are desired, then one measurement suffices since two angles are complementary.

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Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

Math is Fun: Right Angled Triangles
Plain-language definition of the sides and angles of a right triangle.
Khan Academy: Trigonometry with right triangles
Lessons on using inverse trig functions to find right-triangle angles.
Wolfram MathWorld: Right Triangle
Reference definitions for right-triangle sides, angles, and relations.
NIST: Trigonometric functions and units
Reference for trigonometric ratios and angle units.